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arxiv: 2606.28272 · v1 · pith:X3UEUY5Dnew · submitted 2026-06-26 · ✦ hep-ph

Neutrino oscillation data and a pseudo-Dirac heavy neutral lepton

Jan Hajer This is my paper

Pith reviewed 2026-06-29 03:12 UTC · model grok-4.3

classification ✦ hep-ph
keywords neutrino oscillationsheavy neutral leptonspseudo-Diracseesaw modelslepton number violationMajorana phaseflavour mixingneutrinoless double beta decay
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The pith

Light neutrino oscillation data fixes an ellipse in the flavour simplex for active-heavy mixing in the minimal pseudo-Dirac HNL model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines symmetry-protected seesaw models that fit light neutrino oscillation data while allowing heavy neutral leptons to remain potentially reachable at colliders. It focuses on the minimal case of exactly one pseudo-Dirac HNL pair, where an approximate lepton number symmetry keeps the light neutrino masses small. The lepton-number conserving limit is diagonalized exactly without expanding in the mixing, after which the small violating entries are added perturbatively. This produces a symmetry-protected reconstruction of the active-heavy interaction matrix. The rank-two light neutrino mass matrix sets the active flavour direction, and after the amplitude and rotation cancel, the normalised weights lie on an ellipse fixed by the oscillation parameters and the Majorana phase.

Core claim

In the minimal pseudo-Dirac HNL scenario, exact diagonalization of the lepton-number conserving Dirac limit followed by perturbative inclusion of the violating entries yields a symmetry-protected flavour reconstruction of the active-heavy interaction matrix. The rank-two light-neutrino mass matrix fixes the normalised active-flavour direction, while the remaining high-energy information reduces to a single complex light-heavy amplitude. For the normalised leading active-heavy interaction weights, this amplitude and the heavy-sector rotation cancel, leaving an ellipse in the flavour simplex determined by light-neutrino oscillation data and the Majorana phase.

What carries the argument

the ellipse in the flavour simplex determined by light-neutrino oscillation data and the Majorana phase, obtained after exact diagonalization of the LN-conserving limit and perturbative treatment of LN-violating terms

If this is right

  • The rank-two light-neutrino mass matrix fixes the normalised active-flavour direction in the interaction matrix.
  • The remaining high-energy information is a single complex light-heavy amplitude whose phase defines a CP-odd light-heavy invariant.
  • Linear LN-violating terms enter coherent heavy-neutrino oscillations.
  • The neutrinoless double beta effective mass receives contributions from the linear LN-violating terms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Collider searches for HNLs could test whether observed mixing ratios fall on the ellipse predicted by oscillation data.
  • A measured value of the Majorana phase would directly constrain the shape and orientation of the ellipse.
  • The cancellation between amplitude and rotation may appear in other symmetry-protected models with additional HNL pairs.

Load-bearing premise

The lepton-number violating entries are small enough to be treated perturbatively after exact diagonalization of the conserving limit, and the model contains exactly one pseudo-Dirac HNL pair.

What would settle it

Observation of a heavy neutral lepton decay whose active-flavour mixing ratios lie outside the ellipse fixed by current neutrino oscillation data and the Majorana phase.

read the original abstract

Symmetry-protected seesaw models can accommodate light-neutrino oscillation data while keeping heavy neutral leptons (HNLs) within collider reach. In these models, the smallness of the light-neutrino masses is protected by an approximate lepton number (LN)-like symmetry that is broken only by small parameters. We study the minimal scenario in which the new states form one pseudo-Dirac HNL pair. The exact LN-conserving Dirac limit is diagonalised without expanding in the active-sterile mixing, and the small LN-violating entries are then included perturbatively. This yields a symmetry-protected flavour reconstruction of the active-heavy interaction matrix. The rank-two light-neutrino mass matrix fixes the normalised active-flavour direction, while the remaining high-energy information is a single complex light-heavy amplitude whose phase defines a CP-odd light-heavy invariant. For the normalised leading active-heavy interaction weights, this amplitude and the heavy-sector rotation cancel, leaving an ellipse in the flavour simplex determined by light-neutrino oscillation data and the Majorana phase. We also identify how the linear LN-violating terms enter coherent heavy-neutrino oscillations and the neutrinoless double beta effective mass.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims that symmetry-protected seesaw models with one pseudo-Dirac HNL pair allow exact diagonalization of the lepton-number conserving Dirac limit, after which small LN-violating entries are added perturbatively. This produces a symmetry-protected flavour reconstruction in which the rank-two light-neutrino mass matrix fixes the normalised active-flavour direction; the remaining information is a single complex light-heavy amplitude whose phase is a CP-odd invariant. For the normalised leading active-heavy interaction weights, this amplitude and the heavy-sector rotation cancel, leaving an ellipse in the flavour simplex fixed solely by light-neutrino oscillation data and the Majorana phase. The work also identifies how linear LN-violating terms enter coherent heavy-neutrino oscillations and the neutrinoless double-beta effective mass.

Significance. If the claimed cancellation survives the perturbative LN-violating corrections without shifting the leading normalised direction, the result supplies a direct, largely parameter-free link between oscillation parameters and HNL flavour structure. This would be useful for collider phenomenology and 0νββ predictions in symmetry-protected seesaws. The exact (non-expanded) diagonalization of the conserving limit is a methodological strength that avoids uncontrolled approximations in the active-sterile mixing.

major comments (1)
  1. [flavour reconstruction / perturbative LN-violating block] The central ellipse result (abstract and the flavour-reconstruction section) rests on the statement that, after exact LN-conserving diagonalization, the small LN-violating block does not modify the leading normalised active-heavy weights at the same order as the claimed cancellation. This assumption is load-bearing: if O(ε) corrections shift the direction of the normalised mixing vector, the amplitude-rotation cancellation would no longer leave an ellipse determined only by external oscillation data. An explicit order-by-order expansion of the normalised weights (or a symmetry argument showing the corrections vanish) is required to substantiate the claim.
minor comments (2)
  1. The abstract refers to 'the remaining high-energy information is a single complex light-heavy amplitude'; an explicit definition of this amplitude in terms of the mixing-matrix elements, together with its relation to the Majorana phase, would improve readability.
  2. The restriction to exactly one pseudo-Dirac pair is stated as part of the minimal scenario; a brief remark on why the rank-two light mass matrix would not fix a unique direction for two or more pairs would clarify the scope.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a load-bearing assumption in the flavour-reconstruction argument. We address the single major comment below.

read point-by-point responses

  1. Referee: [flavour reconstruction / perturbative LN-violating block] The central ellipse result (abstract and the flavour-reconstruction section) rests on the statement that, after exact LN-conserving diagonalization, the small LN-violating block does not modify the leading normalised active-heavy weights at the same order as the claimed cancellation. This assumption is load-bearing: if O(ε) corrections shift the direction of the normalised mixing vector, the amplitude-rotation cancellation would no longer leave an ellipse determined only by external oscillation data. An explicit order-by-order expansion of the normalised weights (or a symmetry argument showing the corrections vanish) is required to substantiate the claim.

    Authors: We agree that an explicit verification is required. The symmetry-protected construction fixes the leading normalised active-heavy direction in the exact LN-conserving limit; the subsequent perturbative inclusion of the LN-violating block modifies the overall mixing matrix at O(ε), but the normalised vector itself receives corrections only at O(ε²) because the leading eigenvector is protected by the rank-two light-neutrino mass matrix and the residual symmetry. Nevertheless, to make this transparent we will add an appendix containing the order-by-order expansion of the normalised weights through O(ε). This will confirm that the amplitude-rotation cancellation remains intact at the order relevant for the ellipse and that the result is determined solely by oscillation data and the Majorana phase. revision: yes

Circularity Check

0 steps flagged

No significant circularity; ellipse fixed by external oscillation data after exact LN-conserving diagonalization

full rationale

The derivation diagonalizes the LN-conserving limit exactly, then adds small LN-violating terms perturbatively. The central claim states that the normalised active-heavy weights are fixed by the rank-two light mass matrix from oscillation data plus the Majorana phase, with amplitude and rotation cancelling. No quoted equation reduces this output to a fitted parameter or self-citation by construction. The restriction to one pseudo-Dirac pair is an explicit model choice, not a hidden redefinition. Self-citation load-bearing is absent from the provided abstract and reader's summary. This is the normal case of an independent derivation from external data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the claim rests on the domain assumption of an approximate lepton number symmetry broken by small parameters and the perturbative validity of that breaking.

axioms (1)
  • domain assumption Approximate lepton number symmetry protects light neutrino masses and is broken only by small parameters
    Stated as the foundation of symmetry-protected seesaw models in the abstract.

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discussion (0)

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